{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:KUNJUDK6DIZK4IUB3IIMPTQFMG","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6d77f145d5d90881adbd3002f30d0a730be752f944ff57914d607eedb525e1e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-10-21T15:24:53Z","title_canon_sha256":"9f354fb2779235957b4f1097cac40c4f86e21765bf6236bcc25157a44970ff67"},"schema_version":"1.0","source":{"id":"1310.5598","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1310.5598","created_at":"2026-05-18T03:09:32Z"},{"alias_kind":"arxiv_version","alias_value":"1310.5598v2","created_at":"2026-05-18T03:09:32Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1310.5598","created_at":"2026-05-18T03:09:32Z"},{"alias_kind":"pith_short_12","alias_value":"KUNJUDK6DIZK","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"KUNJUDK6DIZK4IUB","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"KUNJUDK6","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:a2666e0c1f8e7d71bfc6ec60a510e8d0804d36b87ddefead667f4f5b8529b99e","target":"graph","created_at":"2026-05-18T03:09:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this short note we prove that the projective dimension of a sequentially Cohen-Macaulay square-free monomial ideal is equal to the maximal height of its minimal primes (also known as the big height), or equivalently, the maximal cardinality of a minimal vertex cover of its facet complex. This in particular gives a formula for the projective dimension of facet ideals of these classes of ideals, which are known to be sequentially Cohen-Macaulay: graph trees and simplicial trees, chordal graphs and some cycles, chordal clutters and graphs, and some path ideals to mention a few. Since polarizat","authors_text":"Sara Faridi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-10-21T15:24:53Z","title":"The projective dimension of sequentially Cohen-Macaulay monomial ideals"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.5598","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:258fddab4ae2a571d229a780fb5d4099066f626f2e0930869412eb326e384eb9","target":"record","created_at":"2026-05-18T03:09:32Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6d77f145d5d90881adbd3002f30d0a730be752f944ff57914d607eedb525e1e5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2013-10-21T15:24:53Z","title_canon_sha256":"9f354fb2779235957b4f1097cac40c4f86e21765bf6236bcc25157a44970ff67"},"schema_version":"1.0","source":{"id":"1310.5598","kind":"arxiv","version":2}},"canonical_sha256":"551a9a0d5e1a32ae2281da10c7ce0561af1768e44dd71122f2d25d75a915e62b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"551a9a0d5e1a32ae2281da10c7ce0561af1768e44dd71122f2d25d75a915e62b","first_computed_at":"2026-05-18T03:09:32.689213Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:32.689213Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jl6QUL+Pf9QNpsJXSgXTe053fbP6r/RoqWrK68tFf5PVbvoHn/QT9kirZHDloYYejwLPgF0s32xmNcybjmFHBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:32.689692Z","signed_message":"canonical_sha256_bytes"},"source_id":"1310.5598","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:258fddab4ae2a571d229a780fb5d4099066f626f2e0930869412eb326e384eb9","sha256:a2666e0c1f8e7d71bfc6ec60a510e8d0804d36b87ddefead667f4f5b8529b99e"],"state_sha256":"bbf27a273739a1938f986de955e4aaf8f13bc72b29085bb77d69fedab1b893b5"}